Statistical techniques for ordinal data

© aSup-2007
STATISTICAL TECHNIQUES FOR ORDINAL DATA   
1
STATISTICAL TECHNIQUES
FOR ORDINAL DATA

© aSup-2007
2
TOOLS WILL NEED
 Ordinal scales
 Probability (the unit normal table)
 Introduction to hypothesis testing
 Correlation

© aSup-2007
3
Preview
 People have a passion for rankings things
Nile is the longest river in the world
The Labrador retriever is the number one
registered dog in the US
English is the fourth common native
language in the world (after Chinese, Hindi,
and Spanish)
Universitas Indonesia is number 201 top
university in the world (THES, 2009)

© aSup-2007
4
Preview
 Part of fascination with rank is that they are
easy to obtain and they are easy to
understand
What is your favorite ice-cream flavor?
 Ordinal scales are less demanding and less
sophisticated than the interval or ratio
scales  easier to use
 ordinal scales can cause some problems
for statistical analysis

© aSup-2007
5
Preview
 Because ordinal data (ranks) provide
limited information they must be used and
interpreted carefully
 Standard statistical methods such as means,
t test, or analysis F variance should not be
used when data are measured on an ordinal
scale

© aSup-2007
6
DATA FROM AN ORDINAL SCALE
 Ordinal values (ranks) only tell the
direction from one score to another, but
provide no information about the distance
between scores
 In a horse race, for example, we know that
the second-place horse is somewhere
between the first- and the third-place horses
 a rank of second is not necessarily
halfway between first and third

© aSup-2007
7
OBTAINING ORDINAL MEASUREMENT
1. Ranks are simpler.
“He is little taller than I am”
2. The original score may violate some of the
basic assumption that underlie certain
statistical procedures.
 the homogeneity of variance assumption
3. The original score may have unusually high
variance
4. Occasionally, an experiment produce
undetermined, or infinite, score

© aSup-2007
8
 Rank the following scores
3 4 4 7 9 9 9 12
14 3 4 0 3 5 14 3
Boy’s score 8, 17, 14, 21
Girl’s score 18, 25, 23, 21, 34, 28, 32, 30, 13
LEARNING CHECK

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9
THE MANN-WHITNEY U TEST
An Alternative to
The Independent-Measures t Test

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10
… is designed to use the data from two
separate samples to evaluate the difference
between two treatment (or two population)
 The calculations for this test require that the
individual scores in the two samples

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11
 A real difference between the two
treatments should cause the scores in one
sample to be generally larger than the score
in the other sample
 If the two sample are combined and all the
scores placed in rank order on a line, the
scores from one sample should be
concentrated at one end of the line, and the
scores from the other sample should be
concentrated at the other end

© aSup-2007
12
The scores from the two samples are clustered
at opposite ends of the rank ordering
1 2 173 4 5 6 14 167 8 9 10 11 12 1513 18
In this case, the data suggest a systematic
difference between the two treatment (or
two samples)
Sample from
treatment A
Sample from
treatment B

© aSup-2007
13
 On the other hand, if there is no treatment
difference, the large and small scores will be
mixed evenly in the two samples because
there is no reason for one set of scores to be
systematically larger or smaller than the
other

© aSup-2007
14
The scores from the two samples are
intermixed evenly along the scale
1 2 173 4 5 6 14 167 8 9 10 11 12 1513 18
In this case, the data indicating no
consistent difference between the two
treatment (or two samples)
Sample from
treatment A
Sample from
treatment B

© aSup-2007
15
THE NULL HYPOTHESIS FOR
 Because the Mann-Whitney test compares
two distributions (rather than two means),
the hypotheses tend to be somewhat vague
H0 : There is no difference between treatments
 therefore, there is no tendency for the ranks
in one treatment condition to be systematically
higher (or lower) than the ranks in the other
treatment condition

© aSup-2007
16
CALCULATION OF THE MANN-WHITNEY U
Sample A : 27, 2, 9, 48, 6, 15
Sample B : 71, 63, 18, 68, 94, 8
 Combine the two samples and all 12 scores
are placed in rank order
2, 6, 8, 9, 15, 18, 27, 48, 63, 68, 71, 94
 Each individual in sample A is assigned 1
point every score in sample B that has a
higher rank

© aSup-2007
17
CALCULATION OF THE MANN-WHITNEY U
RANK
ORDERED SCORES
POINTS FOR
SAMPLE A
POINTS FOR
SAMPLE BSCORE SAMPLE
1 2 A 6 points
2 6 A 6 points
3 8 B 4 points
4 9 A 5 points
5 15 A 5 points
6 18 B 2 points
7 27 A 4 points
8 48 A 4 points
9 63 B 0 points
10 68 B 0 points
11 71 B 0 points
12 94 B 0 points
UA + UB =
nAnB
30 + 6 =
6(6)

© aSup-2007
18
RANKING WITHOUT SCORES
 We assumed we had obtained a score for
each individual. However, it is not
necessary to have a set previously obtained
scores
For example, a researcher could observe a
group of 12 preschool children (6 boys and
6 girls) and rank them in terms aggressive
behavior

© aSup-2007
19
COMPUTING U FOR LARGE
SAMPLES
RANK
ORDERED SCORES
SCORE SAMPLE
1 2 A
2 6 A
3 8 B
4 9 A
5 15 A
6 18 B
7 27 A
8 48 A
9 63 B
10 68 B
11 71 B
12 94 B
UA = nAnB +
nA(nA+1)
2
- Σ RA
UB = nAnB +
nB(nB+1)
2
- Σ RB

© aSup-2007
20
THE MANN-WHITNEY U
… is the smaller U
See Table B.9A
To be significant for any given nA and nB,
the obtained U must be equal to or less
than the critical value in the table.

© aSup-2007
Reporting in Literature
The original scores measured in
questionaire score, were rank-ordered and
a Mann-Whitney U-test was used to
compared the ranks for the n=6 group A
versus n=6 group B. The results indicate
there is significant difference between
group A and group B, U=6, p<.05 one-
tailed with the sum of the ranks equal to
27 for group A and 61 for group B.
21

© aSup-2007
22
HYPOTHESIS TESTS WITH
THE MANN-WHITNEY U
 A large difference between the two
treatments (or samples) causes all the ranks
from sample A to cluster at one end of the
scale all the ranks from sample B to cluster
at the other.
 At the extreme, there is no overlap between
two sample  the Mann-Whitney U will be
zero because one of the sample gets no
point at all

© aSup-2007
23
Psychosis such as schizophrenia often expressed in
the artistic work produced by patients. To test the
reliability of this phenomenon, a psychologist
collected 10 painting done by schizophrenic patient
and another 10 painting by normal college student.
A professor in art department was asked to rank
order all 20 paintings in term bizarreness.
Schizophrenic patents: 1, 3, 4, 5, 6, 8, 9, 11, 12, 14
Student: 2, 7, 10, 13, 15, 16, 17, 18, 19, 20
Test at the .01 level of significance
LEARNING CHECK

© aSup-2007
24
The KRUSKAL-WALLIS Test
An Alternative to
The Independent-Measures ANOVA

© aSup-2007
Kruskal-Wallis Test
Berbeda dengan analisis Mann-Whitney U
Test yang terbatas untuk membandingkan
2 kelompok (treatment) yang terpisah,
analisis Kruskal-Wallis T Test digunakan
untuk mengevaluasi perbedaan urutan
individu dari 3 kelompok atau lebih yang
independen (between subjects)
25

© aSup-2007
26
Ilustrasi Penelitian
Seorang peneliti ingin mengetahui pengaruh
kelembapan (kadar air) udara terhadap
performa kerja karyawan dalam mengetik.
3(tiga) kelompok karyawan dipilih secara
acak untuk ditempatkan pada 3(tiga) ruangan
secara terpisah. Ketiga ruangan tersebut
diatur agar memiliki kelembapan udara
rendah (60%), sedang (75%), dan tinggi (90%).
Kecepatan mengetik diukur dengan urutan
menyelesaikan mengetik ulang suatu tulisan
yang diberikan peneliti.

© aSup-2007
Kruskal-Wallis Test
 Apabila hasil pengukuran (original data) DV
memiliki varians yang besar dan terdapat
undetermined/infinite score, maka akan lebih
tepat menggunakan analisis Kruskal-Wallis
dibandingkan dengan One-Way ANOVA
 Skala pengukuran DV merupakan skala
ordinal (rank-order) atau data numerik
(skala interval/rasio) diubah dalam bentuk
rank-order (ordinal)
27

© aSup-2007
Kruskal-Wallis Test
Tujuan penelitiannya adalah untuk
mengetahui apakah kelompok treatment yang
satu akan memiliki ranking yang secara
konsisten lebih tinggi (atau lebih rendah)
dibandingkan kelompok lainnya.
28

© aSup-2007
Hipotesis Kruskal-Wallis Test
 H0: tidak ada kecenderungan bahwa ranking
pada kelompok treatment tertentu akan
secara sistematis lebih tinggi (atau lebih
rendah) dibandingkan ranking pada
kelompok treatment yang lain.
Dengan demikian, tidak ada perbedaan
yang signifikan di antara kelompok
treatment
 HA: sedikitnya ranking pada satu kelompok treatment secara
sistematis lebih tinggi (atau lebih rendah) dibandingkan
ranking pada kelompok treatment yang lain. 29

© aSup-2007
 H0: tidak ada kecenderungan bahwa ranking pada kelompok
treatment tertentu akan secara sistematis lebih tinggi (atau
lebih rendah) dibandingkan ranking pada kelompok treatment
yang lain.
Dengan demikian, tidak ada perbedaan yang signifikan di
antara kelompok treatment
 HA: sedikitnya ranking pada satu kelompok
treatment secara sistematis lebih tinggi (atau
lebih rendah) dibandingkan ranking pada
kelompok treatment yang lain.
Dengan demikian, terdapat perbedaan yang
signifikan di antara kelompok treatment. 30

© aSup-2007
Pengaruh Kelembapan terhadap
Kecepatan Mengetik (menit:detik)
Kelembapan Udara
Rendah Sedang Tinggi
5:07 4:12 8:25
4:28 5:07 4:40
6:39 4:49 5:07
4:35 4:58 5:43
5:16 4:35 6:14
31
Kelembapan Udara
Rank?

© aSup-2007
Kelembapan Udara
5:07 4:12 8:25
4:28 5:07 4:40
6:39 4:49 5:07
4:35 4:58 5:43
5:16 4:35 6:14
32
Kelembapan Udara
1

© aSup-2007
Kelembapan Udara
5:07 4:12 8:25
4:28 5:07 4:40
6:39 4:49 5:07
4:35 4:58 5:43
5:16 4:35 6:14
33
Kelembapan Udara
1
2

© aSup-2007
Kelembapan Udara
5:07 4:12 8:25
4:28 5:07 4:40
6:39 4:49 5:07
4:35 4:58 5:43
5:16 4:35 6:14
34
Kelembapan Udara
1
2
3,5
3,5

© aSup-2007
Kelembapan Udara
5:07 4:12 8:25
4:28 5:07 4:40
6:39 4:49 5:07
4:35 4:58 5:43
5:16 4:35 6:14
35
Kelembapan Udara
1
2 5
3,5
3,5

© aSup-2007
Kelembapan Udara
5:07 4:12 8:25
4:28 5:07 4:40
6:39 4:49 5:07
4:35 4:58 5:43
5:16 4:35 6:14
36
Kelembapan Udara
1
2 5
6
3,5
3,5

© aSup-2007
Kelembapan Udara
5:07 4:12 8:25
4:28 5:07 4:40
6:39 4:49 5:07
4:35 4:58 5:43
5:16 4:35 6:14
37
Kelembapan Udara
1
2 5
6
3,5 7
3,5

© aSup-2007
Kelembapan Udara
5:07 4:12 8:25
4:28 5:07 4:40
6:39 4:49 5:07
4:35 4:58 5:43
5:16 4:35 6:14
38
Kelembapan Udara
9 1
2 9 5
6 9
3,5 7
3,5

© aSup-2007
Kelembapan Udara
5:07 4:12 8:25
4:28 5:07 4:40
6:39 4:49 5:07
4:35 4:58 5:43
5:16 4:35 6:14
39
Kelembapan Udara
9 1
2 9 5
6 9
3,5 7
11 3,5

© aSup-2007
Kelembapan Udara
5:07 4:12 8:25
4:28 5:07 4:40
6:39 4:49 5:07
4:35 4:58 5:43
5:16 4:35 6:14
40
Kelembapan Udara
9 1
2 9 5
6 9
3,5 7 12
11 3,5

© aSup-2007
Kelembapan Udara
5:07 4:12 8:25
4:28 5:07 4:40
6:39 4:49 5:07
4:35 4:58 5:43
5:16 4:35 6:14
41
Kelembapan Udara
9 1
2 9 5
6 9
3,5 7 12
11 3,5 13

© aSup-2007
Kelembapan Udara
5:07 4:12 8:25
4:28 5:07 4:40
6:39 4:49 5:07
4:35 4:58 5:43
5:16 4:35 6:14
42
Kelembapan Udara
9 1
2 9 5
14 6 9
3,5 7 12
11 3,5 13

© aSup-2007
Kelembapan Udara
5:07 4:12 8:25
4:28 5:07 4:40
6:39 4:49 5:07
4:35 4:58 5:43
5:16 4:35 6:14
43
Kelembapan Udara
9 1 15
2 9 5
14 6 9
3,5 7 12
11 3,5 13

© aSup-2007
Analisis Kruskal-Wallis T Test
44
Kelembapan Udara
9 1 15
2 9 5
14 6 9
3,5 7 12
11 3,5 13
T1 = 39,5 T2 = 26,5 T3 = 54
n = 5 n = 5 n = 5

© aSup-2007
45
Kelembapan Udara
T1 = 39,5 T2 = 26,5 T3 = 54
n = 5 n = 5 n = 5
T =
12
N(N+1
)
Σ
T2
n
-
3(N+1)
T =
12
15(16)
+
39,52
5
- 3(16)+
26,52
5
542
5

© aSup-2007
T = 3,785 Signifikan?
Table B.8 Chi Square; df = k-1
df = k-1 = 2; critical value = 5,99
3,785 < 5,99  TIDAK Signifikan
 Tidak Ada pengaruh kelembapan
terhadap kecepatan mengetik

© aSup-2007
Kruskal-Wallis Test
Berbeda dengan analisis Wilcoxon T Test
yang terbatas untuk membandingkan 2
kelompok (treatment) yang terpisah,
analisis Friedman Test digunakan untuk
mengevaluasi perbedaan urutan individu
dari 3 treatment atau lebih dari satu
kelompok (within subjects)
48

© aSup-2007
49
Ilustrasi Penelitian
Tiara ingin melihat pengaruh pemberian
pelatihan Empati terhadap keterampilan
berkomunikasi karyawan.
Seorang atasan diminta untuk meranking
keterampilan berkomunikasi setiap karyawan
pada ketiga pengukuran.
Pengukuran keterampilan berkomunikasi
dilakukan sebelum pemberian traning, 3
bulan setelah training, dan 6 bulan setelah
training.

© aSup-2007
Friedman Test
 Apabila hasil pengukuran (original data) DV
memiliki varians yang besar dan terdapat
undetermined/infinite score, maka akan lebih
tepat menggunakan analisis Friedman Test
dibandingkan dengan Repeated-Measures
ANOVA
 Skala pengukuran DV merupakan skala
ordinal (rank-order) atau data numerik
(skala interval/rasio) diubah dalam bentuk
rank-order (ordinal)
50

© aSup-2007
Friedman Test
Tujuan penelitiannya adalah untuk
mengetahui apakah treatment yang satu akan
memiliki ranking yang secara konsisten lebih
tinggi (atau lebih rendah) dibandingkan
treatment lainnya.
51

© aSup-2007
 H0: tidak ada kecenderungan bahwa ranking
pada treatment tertentu akan secara
sistematis lebih tinggi (atau lebih rendah)
dibandingkan ranking pada treatment yang
lain.
Dengan demikian, tidak ada perbedaan
yang signifikan di antara treatment
 HA: sedikitnya ranking pada satu treatment secara sistematis
lebih tinggi (atau lebih rendah) dibandingkan ranking pada
treatment yang lain.
Dengan demikian, terdapat perbedaan yang signifikan di
antara treatment. 52

© aSup-2007
 H0: tidak ada kecenderungan bahwa ranking pada treatment
tertentu akan secara sistematis lebih tinggi (atau lebih rendah)
dibandingkan ranking pada treatment yang lain.
Dengan demikian, tidak ada perbedaan yang signifikan di
antara treatment
 HA: sedikitnya ranking pada satu treatment
secara sistematis lebih tinggi (atau lebih
rendah) dibandingkan ranking pada
treatment yang lain.
Dengan demikian, terdapat perbedaan yang
signifikan di antara treatment. 53

© aSup-2007
Skor Keterampilan Berkomunikasi
Karyawan Sebelum 3-bulan 6-bulan
A 24 22 30
B 19 25 28
C 22 34 30
D 25 28 34
E 20 29 29
F 26 24 33
54

© aSup-2007
Analisis Friedman Test
57
Keterampilan Berkomunikasi
R1 = 8 R2 = 11,5 R3 = 16,5
n = 6 n = 6 n = 6
χ2
r=
12
nk(k+1)
ΣT2
- 3n(k+1)
H =
12
(6)(3)(4)
(8)2
+ (11,5)2
+ (16,5)2 - (3)(6)(4)

© aSup-2007
Analisis Friedman Test
χ2
r= 6,08 Signifikan?
Table B.8 Chi Square; df = k-1
df = 2; critical value = 5,99
6,08 > 5,99  SIGNIFIKAN
 Ada pengaruh pelatihan terhadap
keterampilan berkomunikasi

Statistical techniques for ordinal data

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